TITLE: composition of covering maps
QUESTION [19 upvotes]: The origin of my question arose from a problem: Let $q: X \to Y$ and $r: Y \to Z$ be covering maps, let $p= r \circ q$. Show that if $r^{-1}(z)$ is finite for each $z \in Z$, then $p$ is a covering map.
I am just wondering is there an easy counter example to show the necessary of "inverse image of every point is finite" condition?
 I have already checked Hatcher's book (the wedged circle example) but I wonder is there a less complex and more obvious example?
I tried to construct a counter example by composition of a covering map from $R_+$ to $S^1$ with $R$ in the middle, but I failed due to I can't find a covering map from $R^+$ to $R$ which does the job.

REPLY [5 votes]: You could also look at the paper 
J. Brazas, "Semicoverings: a generalisation of covering space theory", Homology, Homotopy and Applications, Vol. 14 (2012), No. 1, pp.33-63. 
which gives a generalisation of covering maps, so that a covering map is a semicovering, but semicoverings, unlike coverings,  satisfy the "2 in 3" rule: if $h=f\circ g$ and two of $f,g,h$ are semicoverings, then so is the third.